Relative Computational Power Research Articles

EXPANDING THE REALS BY CONTINUOUS FUNCTIONS ADDS NO COMPUTATIONAL POWER

AbstractWe study the relative computational power of structures related to the ordered field of reals, specifically using the notion of generic Muchnik reducibility. We show that any expansion of the reals by a continuous function has no more computing power than the reals, answering a question of Igusa, Knight, and Schweber [7]. On the other hand, we show that there is a certain Borel expansion of the reals that is strictly more powerful than the reals and such that any Borel quotient of the reals reduces to it.

Optimal Design of Bitcoin Mining

Bitcoin is a peer-to-peer electronic currency without central bank controlling. Nowadays, increasing amount of people are engaged in the mining of Bitcoin for great profits. However, in the Bitcoin system, the more participants in the system, the greater computation power of the whole network and the less efficiency in the output of the coin, since large computational power of the whole Bitcoin system will lead to increased difficulty for a single miner to mine a new data block. At the same time, when more than 51% computing power is controlled by a single node, it could destroy the Bitcoin system. In order to reduce ineffective mining behaviors, one would wish to employ the optimal selection mechanism of different miners. This paper will mainly develop the model in which small miners (those who have relative smaller computational power in a block mining) join the major ones based on revenue, computational power cost, and other elements of the process comparing to the current model (when they work separately).

The computability of relaxed data structures: queues and stacks as examples

Most concurrent data structures being designed today are versions of known sequential data structures. However, in various cases it makes sense to relax the semantics of traditional concurrent data structures in order to get simpler and possibly more efficient and scalable implementations. For example, when solving the classical producer-consumer problem by implementing a concurrent queue, it might be enough to allow the dequeue operation (by a consumer) to return and remove one of the two oldest values in the queue, and not necessarily the oldest one. We define infinitely many possible relaxations of several traditional data structures and objects: queues, stacks, multisets and registers, and examine their relative computational power.

On the computational power of WECPAR

Reconfigurable models were shown to be very powerful in solving many problems faster than non-reconfigurable models. WECPAR $$W(M,N,k)$$ W ( M , N , k ) is an $$M\times N$$ M × N reconfigurable model that has point-to-point reconfigurable interconnection with $$k$$ k wires between neighboring processors. This paper studies several aspects of WECPAR. We first consider solving the list ranking problem on WECPAR. Some of the results obtained show that, ranking one element in a list of $$N$$ N elements can be solved on $$W(N,N,N)$$ W ( N , N , N ) WECPAR in $$O(1)$$ O ( 1 ) time. Also, on $$W(N,N,k)$$ W ( N , N , k ) , ranking a list $$L(N)$$ L ( N ) of $$N$$ N elements can be done in $$O((\log N)(\lceil {\log _{k+1} N} \rceil ))$$ O ( ( log N ) ( ? log k + 1 N ? ) ) time. Then, we assess the relative computational power of WECPAR and transfer a large body of algorithms to work directly on WECPAR. We introduce several simulation algorithms between WECPAR and well-known models such as PRAM and RMBM. Simulation algorithms show that a PRIORITY CRCW PRAM $$P(N,S)$$ P ( N , S ) of $$N$$ N processors and $$S$$ S shared memory locations can be simulated on $$W(S,N,k)$$ W ( S , N , k ) WECPAR in $$O(\lceil {\log _{k+1} N}\rceil +\lceil {\log _{k+1} S}\rceil )$$ O ( ? log k + 1 N ? + ? log k + 1 S ? ) time. Also, we show that a PRIORITY CRCW basic RMBM( $$P,B)$$ P , B ) , of $$P$$ P processors and $$B$$ B buses can be simulated on $$W(B, P+B, k)$$ W ( B , P + B , k ) WECPAR in $$O(\lceil {\log _{k+1} (P+B)}\rceil )$$ O ( ? log k + 1 ( P + B ) ? ) time. This directly migrate a large number of algorithms to work on WECPAR with the simulation overhead.

Locality and checkability in wait-free computing

This paper studies notions of locality that are inherent to the specification of distributed tasks by identifying fundamental relationships between the various scales of computation, from the individual process to the whole system. A locality property called projection-closed is identified. This property completely characterizes tasks that are wait-free checkable, where a task $$T =(\mathcal{I },\mathcal{O },\varDelta )$$ is said to be checkable if there exists a distributed algorithm that, given $$s\in \mathcal{I }$$ and $$t\in \mathcal{O }$$ , determines whether $$t\in \varDelta {(s)}$$ , i.e., whether $$t$$ is a valid output for $$s$$ according to the specification of $$T$$ . Projection-closed tasks are proved to form a rich class of tasks. In particular, determining whether a projection-closed task is wait-free solvable is shown to be undecidable. A stronger notion of locality is identified by considering tasks whose outputs “look identical” to the inputs at every process: a task $$T= (\mathcal{I },\mathcal{O },\varDelta )$$ is said to be locality-preserving if $$\mathcal{O }$$ is a covering complex of $$\mathcal{I }$$ . We show that this topological property yields obstacles for wait-free solvability different in nature from the classical impossibility results. On the other hand, locality-preserving tasks are projection-closed, and thus they are wait-free checkable. A classification of locality-preserving tasks in term of their relative computational power is provided. This is achieved by defining a correspondence between subgroups of the edgepath group of an input complex and locality-preserving tasks. This correspondence enables to demonstrate the existence of hierarchies of locality-preserving tasks, each one containing, at the top, the universal task (induced by the universal covering complex), and, at the bottom, the trivial identity task.

A Framework for Heterotic Computing

Computational devices combining two or more different parts, one controlling the operation of the other, for example, derive their power from the interaction, in addition to the capabilities of the parts. Non-classical computation has tended to consider only single computational models: neural, analog, quantum, chemical, biological, neglecting to account for the contribution from the experimental controls. In this position paper, we propose a framework suitable for analysing combined computational models, from abstract theory to practical programming tools. Focusing on the simplest example of one system controlled by another through a sequence of operations in which only one system is active at a time, the output from one system becomes the input to the other for the next step, and vice versa. We outline the categorical machinery required for handling diverse computational systems in such combinations, with their interactions explicitly accounted for. Drawing on prior work in refinement and retrenchment, we suggest an appropriate framework for developing programming tools from the categorical framework. We place this work in the context of two contrasting concepts of "efficiency": theoretical comparisons to determine the relative computational power do not always reflect the practical comparison of real resources for a finite-sized computational task, especially when the inputs include (approximations of) real numbers. Finally we outline the limitations of our simple model, and identify some of the extensions that will be required to treat more complex interacting computational systems.

The computable Lipschitz degrees of computably enumerable sets are not dense

The computable Lipschitz reducibility was introduced by Downey, Hirschfeldt and LaForte under the name of strong weak truth-table reducibility (Downey et al. (2004) [6]). This reducibility measures both the relative randomness and the relative computational power of real numbers. This paper proves that the computable Lipschitz degrees of computably enumerable sets are not dense. An immediate corollary is that the Solovay degrees of strongly c.e. reals are not dense. There are similarities to Barmpalias and Lewis’ proof that the identity bounded Turing degrees of c.e. sets are not dense (George Barmpalias, Andrew E.M. Lewis (2006) [2]), however the problem for the computable Lipschitz degrees is more complex.

Relations Between Several Parallel Computational Models

We investigate the relative computational power of parallel models with shared memory. Based on feasibility considerations present in the literature, we split these models into lightweight and heavyweight, and then find that the heavyweight class is strictly more powerful than the lightweight class, as expected. On the other hand, we contradict the long held belief that the heavyweight models (namely, the Combining CRCW PRAM and the BSR) form a hierarchy, showing that they are identical in computational power with each other. We thus introduce the BSR into the family of practically meaningful massively parallel models. We also investigate the power of concurrent-write on models with reconfigurable buses, finding that it does not add computational power over exclusive-write under certain reasonable assumptions. Overall, the Combining CRCW PRAM and the CREW models with directed reconfigurable buses are found to be the simplest of the heavyweight models, which now also include the BSR and all the models with directed reconfigurable buses. These results also have significant implications in the area of real-time computations.

Computation in finitary stochastic and quantum processes

We introduce stochastic and quantum finite-state transducers as computation-theoretic models of classical stochastic and quantum finitary processes. Formal process languages, representing the distribution over a process’ behaviors, are recognized and generated by suitable specializations. We characterize and compare deterministic and nondeterministic versions, summarizing their relative computational power in a hierarchy of finitary process languages. Quantum finite-state transducers and generators are a first step toward a computation-theoretic analysis of individual, repeatedly measured quantum dynamical systems. They are explored via several physical systems, including an iterated-beam-splitter, an atom in a magnetic field, and atoms in an ion trap—a special case of which implements the Deutsch quantum algorithm. We show that these systems’ behaviors, and so their information processing capacity, depends sensitively on the measurement protocol.

A hypercomputational alien

Is there a physical constant with the value of the halting function? An answer to this question, as in other discussions of hypercomputation, assumes a fixed interpretation of nature by mathematical entities. Without agreeing on such an interpretation, the question is without context and meaningless. We discuss the subjectiveness of viewing the mathematical properties of nature, and the possibility of comparing computational models having alternate views of the world. For that purpose, we propose a conceptual framework for power comparison, by linking computational models to hypothetical physical devices. Accordingly, we deduce some mathematical notions of relative computational power, allowing for the comparison of arbitrary models over arbitrary domains. In addition, we demonstrate that the method commonly used in the literature for establishing that one model is strictly more powerful than another is problematic, as it can allow for a model to be “more powerful” than itself. On the positive side, we note that Turing machines and the recursive functions are not susceptible to this anomaly, justifying the standard means of showing that a model is more powerful than Turing machines.

Logspace Reducibility: Models and Equivalences

We study the relative computational power of logspace reduction models. In particular, we study the relationships between one-way and two-way oracle tapes, resetting of the oracle head, and blanking of the oracle tape. We show that oracle models letting information persist between queries can be quite powerful, even if the information is not readable by the querying machine. We show that logspace f(n)-Turing reductions are stronger than polynomial-time f(n)-Turing reductions when f(n) = ω(log n), and that this is optimal if P = L.

Modelling the complexity of parallel and VLSI computations with Boolean circuits

Complexity theory seeks to understand the resource requirements inherent in the solution of problems on computers. It also seeks to understand the relative computational power of different models. The Turing model is difficult to work with, in part because of the impossibility of studying strictly finite structures. Boolean circuits are playing a more important role in our understanding of computation. They are useful as models in situations as far removed as VLSI design and parallel computation. The branching program challenges our intuitions. It has been shown in an indirect fashion that constant-width branching programs can ‘count’, but not exactly how. Algebraic methods give a handle and allow us for the first time to study complicated combinatorial structures.